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If a real number can be written as the product of natural numbers raised to rational exponents, there is indeed a unique way to express it as a product of distinct primes raised to rational exponents. This follows from the unique factorization theorem for integers (as has been said in the comments) through elementary if slightly messy arguments. So I think the person who removed your "prime factorizations" from the pitch intervals page, on the ground that "(prime) factors not meaningful for non-rational intervals", is being pedantic. Whether or not the removed expressions (which can be easily derived from the equivalent ones that were left in the adjacent boxes) were likely to be useful, I don't really know, not being a musician.